Abstract:Low-rank decomposition is a compelling approach for compressing large language models, but its effectiveness hinges on selecting which singular-vector bases to retain for a target task. Existing methods such as Basel adapt singular-value coefficients on downstream data and prune bases with small re-learned magnitudes, a heuristic that can be misaligned with task performance because it ignores the local geometry of the loss landscape. We present Basis Selection with Importance (BSI), a principled low-rank compression framework that ranks and prunes bases by directly estimating the expected loss increase incurred when each basis is removed. BSI derives a derivative-based importance score from a second-order Taylor expansion of the task loss with respect to singular values, combining first-order sensitivity and second-order curvature to quantify pruning impact. To make this criterion practical for LLMs, we develop an efficient Hessian-diagonal estimator by adapting the Hutchinson randomized-probing method to loss curvature with symmetric parameter perturbations. We provide a comprehensive theoretical analysis, including loss-increase bounds under basis pruning, explicit propagation of Hessian-diagonal estimation error into these bounds, variance characterization tied to the Hessian spectrum, high-probability sample-complexity guarantees for achieving a target estimation accuracy, and guidance on perturbation intensity. Extensive experiments on mathematical reasoning benchmarks demonstrate that BSI consistently outperforms state-of-the-art low-rank decomposition baselines, with especially strong improvements under deep compression.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.01627 [cs.LG] |
| (or arXiv:2605.01627v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.01627 arXiv-issued DOI via DataCite (pending registration) |
From: Daniel Asante [view email]
[v1]
Sat, 2 May 2026 22:35:02 UTC (70 KB)
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